Theory of Computing Systems

, Volume 56, Issue 4, pp 630–642 | Cite as

New Lower Bounds on Circuit Size of Multi-output Functions

  • Evgeny Demenkov
  • Alexander S. Kulikov
  • Olga Melanich
  • Ivan Mihajlin
Article

Abstract

Let Bn, m be the set of all Boolean functions from {0, 1}n to {0, 1}m, Bn = Bn, 1 and U2 = B2∖{⊕, ≡}. In this paper, we prove the following two new lower bounds on the circuit size over U2.
  1. 1.
    A lower bound \(C_{U_{2}}(f) \ge 5n-o(n)\) for a linear function fBn − 1,logn. The lower bound follows from the following more general result: for any matrix A ∈ {0, 1}m × n with n pairwise different non-zero columns and b ∈ {0, 1}m,
    $$C_{U_{2}}(Ax \oplus b)\ge 5(n-m).$$
     
  2. 2.
    A lower bound \(C_{U_{2}}(f) \ge 7n-o(n)\) for fBn, n. Again, this is a consequence of the following result: for any fBn satisfying a certain simple property,
    $$C_{U_{2}}(g(f)) \ge \min \{C_{U_{2}}(f|_{x_{i} = a, x_{j} = b}) \colon x_{i} \neq x_{j}, a,b, \in \{0,1\}\} +2n-\varTheta (1)$$
    where g(f) ∈ Bn, n is defined as follows: g(f) = (fx1, … , fxn) (to get a 7no(n) lower bound it remains to plug in a known function fBn, 1 with \(C_{U_{2}}(f) \ge 5n-o(n)\)).
     

Keywords

Circuit complexity Lower bounds Boolean functions Gate elimination 

Notes

Acknowledgements

We would like to thank the anonymous reviewers for their comments that helped us to improve the readability of the paper.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Evgeny Demenkov
    • 1
  • Alexander S. Kulikov
    • 2
  • Olga Melanich
    • 2
  • Ivan Mihajlin
    • 2
  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.St. Petersburg Department of Steklov Institute of MathematicsSt. PetersburgRussia

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